4-1 Lesson 4 Objectives Development of source terms Development of source terms Review of Legendre expansions Review of Legendre expansions Resulting full Boltzmann Equation Resulting full Boltzmann Equation Source vs. Eigenvalue calculations Source vs. Eigenvalue calculations Four eigenvalue formulations Four eigenvalue formulations

4-2 BE so far The time-independent Boltzmann Equation we have derived to this point is: The time-independent Boltzmann Equation we have derived to this point is: But you will remember that we swept a bunch of terms under the rug by wrapping them up into the q term But you will remember that we swept a bunch of terms under the rug by wrapping them up into the q term (Which ones?) (Which ones?)

4-3 Transport with Secondary Particles We will now “unwrap” the source terms: We will now “unwrap” the source terms: External fixed sources External fixed sources Scattering sources Scattering sources Fission sources Fission sources

4-4 External Fixed Source This source term comprises particle sources that do not depend on flux (e.g., radioactive isotopes, cosmic rays) This source term comprises particle sources that do not depend on flux (e.g., radioactive isotopes, cosmic rays) These are simply specified for the calculation as: These are simply specified for the calculation as: In many cases of interest there is no or dependence. In many cases of interest there is no or dependence.

4-5 Use of Legendre expansions Using the cosine of the deflection angle, we can represent the angular dependence of the distribution in a Legendre expansion: Using the cosine of the deflection angle, we can represent the angular dependence of the distribution in a Legendre expansion: This allows us to represent the scattering distribution by determining the Legendre coefficients: This allows us to represent the scattering distribution by determining the Legendre coefficients:

4-6 Use of Legendre expansions (2) Using the orthogonality of the Legendre polynomials: Using the orthogonality of the Legendre polynomials: We can operate on both sides of the expansion (1 st eqn. previous slide) with: We can operate on both sides of the expansion (1 st eqn. previous slide) with:

4-7 Use of Legendre expansions (3) And remembering that a Kronecker delta works pulls out a single element like this: And remembering that a Kronecker delta works pulls out a single element like this: To get: To get: Work this out for yourself (Prob. 4-1) Work this out for yourself (Prob. 4-1)

4-8 Scattering Source This source term comprises all particle reactions (other than fission) from which particles that we are interested in are emitted This source term comprises all particle reactions (other than fission) from which particles that we are interested in are emitted The basic cross section is: The basic cross section is: Note that it is a distribution in destination energy and direction Note that it is a distribution in destination energy and direction

4-9 Scattering Source (2) The scattering source is: The scattering source is: Again, Legendre expansions are normally used for the scattering cross section: Again, Legendre expansions are normally used for the scattering cross section:

4-10 Scattering Source (3) The coefficients are given by: The coefficients are given by: Substituting this expression gives: Substituting this expression gives:

4-11 Scattering Source (4) However, we can use the Legendre addition theorem, which says: However, we can use the Legendre addition theorem, which says:where

4-12 Scattering Source (5) Substituting this expression gives: Substituting this expression gives:where

4-13 Scattering Source (5) Conveniently, this term is also the coefficient of the angular flux expansion in spherical coordinates: Conveniently, this term is also the coefficient of the angular flux expansion in spherical coordinates: The scattering terms are therefore implemented through these flux moments The scattering terms are therefore implemented through these flux moments

4-14 Fission Source The text develops the time-dependent fission source term, including prompt and delayed fission neutron terms. The text develops the time-dependent fission source term, including prompt and delayed fission neutron terms. Our primary concern is not time-dependent, so we will use: Our primary concern is not time-dependent, so we will use: Note that there is no angular dependence and that it is “per unit solid angle” Note that there is no angular dependence and that it is “per unit solid angle”

4-15 Complete Source Combining the three parts of the source: Combining the three parts of the source: where the “double zero” moment is equivalent to the normal scalar flux.

4-16 Full Equation The full time-independent Boltzmann Equation is: The full time-independent Boltzmann Equation is:

4-17 Source vs. Eigenvalue Calculations The nature of the source terms divides the solution into two categories: The nature of the source terms divides the solution into two categories: Source problems: subcritical with: Source problems: subcritical with: Eigenvalue problems with: Eigenvalue problems with: The subcriticality requirement is because there is no time-independent physical solution for critical or super-critical systems with sources The subcriticality requirement is because there is no time-independent physical solution for critical or super-critical systems with sources Mathematical solutions would have negative fluxes Mathematical solutions would have negative fluxes

4-18 Eigenvalue Calculations Without external souces, we get the homogeneous equation: Without external souces, we get the homogeneous equation: Two characteristics of the solution: Two characteristics of the solution: 1. Any constant times a solution is a solution. 2. There probably isn’t a meaningful solution

4-19 Eigenvalue solution normalization For the first point, we generally either normalize to 1 fission neutron: For the first point, we generally either normalize to 1 fission neutron: or to a desired power level: or to a desired power level: where k is a conversion constant (e.g., 200 MeV/fission) where k is a conversion constant (e.g., 200 MeV/fission)

4-20 Eigenvalue approach For the second problem (i.e., no meaningful solution), we deal with it by adding a term with a constant that we can adjust to achieve balance in the equation. For the second problem (i.e., no meaningful solution), we deal with it by adding a term with a constant that we can adjust to achieve balance in the equation. We will discuss four different eigenvalue formulations: We will discuss four different eigenvalue formulations: 1. Lambda (k-effective) eigenvalue 2. Alpha (time-absorption) eigenvalue 3. B 2 (buckling) eigenvalue 4. Material search “eigenvalue”

4-21 Lambda (k-effective) eigenvalue The first (and most common) eigenvalue form involves dividing, the number of neutrons emitted per fission: The first (and most common) eigenvalue form involves dividing, the number of neutrons emitted per fission: Keep largest of multiple eigenvalues Keep largest of multiple eigenvalues

4-22 Lambda eigenvalue (2) The criticality state is given by: The criticality state is given by: Advantages: Advantages: 1. Everybody uses it 2. Guaranteed real solution 3. Fairly intuitive (if you don’t take it too seriously) 4. Good measure of distance from criticality for reactors 5. Very straightforward calculation (no search required) Disadvantages: Disadvantages: 1. No physical basis 2. Not a good measure of distance from criticality for CS

4-23 Alpha (time-absorption) eigenvalue The second eigenvalue form involves adding a term to the removal term: The second eigenvalue form involves adding a term to the removal term: Keep largest of multiple eigenvalues Keep largest of multiple eigenvalues

4-24 Alpha eigenvalue (2) Physical basis is the representation of the time dependence as exponential: Physical basis is the representation of the time dependence as exponential:

4-25 Alpha eigenvalue (3) The criticality state is given by: The criticality state is given by: Advantages: Advantages: 1. Physical basis 2. Intuitive for kinetics work Disadvantages: Disadvantages: 1. No guaranteed real solution 2. Not intuitive for reactor design or CS work 3. Search required (to make k-effective go to 1)

4-26 B 2 (buckling) eigenvalue The third eigenvalue form also involves adding a term to the removal term: The third eigenvalue form also involves adding a term to the removal term: Physical basis is the diffusion theory approximation of leakage by Physical basis is the diffusion theory approximation of leakage by

4-27 B 2 eigenvalue (2) Mathematical basis is the representation of spatial dependence of flux as Fourier transform: Mathematical basis is the representation of spatial dependence of flux as Fourier transform: This substitutes to give us: This substitutes to give us:

4-28 B 2 eigenvalue (3) The criticality state is given by: The criticality state is given by: Advantages: Advantages: 1. Physical basis 2. Good measure of distance from criticality Disadvantages: Disadvantages: 1. No guaranteed real solution 2. Not intuitive for kinetics or CS work 3. Search required (to make k-effective go to 1)

4-29 Homework 4-1 Group constants Group 1 Group 2 Group 3 Group4 1000 ffff aaaa D (cm) V (cm/sec) 2.2e93.2e82.7E72.5e5 Find the lambda, B 2, and alpha eigenvalues for infinite medium cross sections (Use EXCEL, MatLab, whatever) Find the lambda, B 2, and alpha eigenvalues for infinite medium cross sections (Use EXCEL, MatLab, whatever) Find the resulting group fluxes. Find the resulting group fluxes. Use Use Scattering 1->2=.06 2->3=.09 3->4=.6 (others 0) Base eqn: Base eqn: Find the lambda, B 2, and alpha eigenvalues for infinite medium cross sections (Use EXCEL, MatLab, whatever) Find the lambda, B 2, and alpha eigenvalues for infinite medium cross sections (Use EXCEL, MatLab, whatever) Find the resulting group fluxes. Find the resulting group fluxes. Use Use Scattering 1->2=.06 2->3=.09 3->4=.6 (others 0) Base eqn: Base eqn: